Maximum Route Duration Research Articles

An exact branch-and-price-and-cut algorithm for a practical and large-scale dial-a-ride problem

Vehicle routing problems are encountered in various practical transportation scenarios. One such problem is the dial-a-ride problem (DARP), which involves transporting efficiently passengers from their origin to their destination and whose objective is to design vehicle routes minimizing total costs and accommodating all pickup-and-delivery requests. The solution must also adhere to capacity limitations, time windows, and a maximum ride time for each passenger. This paper proposes an exact branch-price-and-cut algorithm to solve a practical variant of the DARP, where both the passengers and vehicle fleet are heterogeneous, and other practical constraints such as break requirements and maximum route duration are imposed. In this algorithm, column generation which alternates between solving a master problem and subproblems, is used to compute lower bounds in the search tree. We develop a labeling algorithm to efficiently handle these subproblems. The effectiveness of the algorithm is evaluated on a real-world case study involving 849 heterogeneous passengers and more than 70 available vehicles, as well as on 10 smaller instances (with 300 to 820 trips) extracted from another very large real-world dataset. To the best of our knowledge, this 849-trip instance is the largest DARP instance reported to be solved to optimality in the literature.

A new formulation and a branch-and-cut algorithm for the set orienteering problem

In this study we address the Set Orienteering Problem, which is a generalization of the Orienteering Problem where customers are clustered in groups. Each group is associated with a profit which is gained in case at least one customer in the group is served. A single vehicle is available to serve the customers. The aim is to find the vehicle route that maximizes the profit collected without exceeding a maximum route cost, which can be interpreted also as route duration. The problem was introduced in Archetti (2018) together with a mathematical programming formulation. In this paper, we propose a new formulation which uses less variables. We also derive different classes of valid inequalities to strengthen the formulation. In addition, separation algorithms are developed, some of which are new with respect to those presented in the literature. A branch-and-cut algorithm is implemented to solve the problem and tests are made on benchmark instances. The results show that the branch-and-cut algorithm is effective in solving instances with up to 100 customers. Moreover, the difficulty of solving the problem largely depends on the maximum route duration. We also show that valid inequalities are effective in speeding up the solution process. Finally, a comparison with two exact benchmark approaches proposed in the literature shows that the branch-and-cut algorithm proposed in this paper is the new state-of-the-art exact approach for solving the Set Orienteering Problem.

Last mile delivery routing problem using autonomous electric vehicles

This paper presents a study on the application of Autonomous Delivery Vehicles (ADVs) in last-mile delivery for urban logistics. Specifically, we focus on a routing problem using multi-stop ADVs, with the goal of minimizing route and vehicle usage costs while satisfying several constraints associated with load and battery capacities, maximum route duration for ADVs, and maximum walking distance for customers. We refer to this problem as the Autonomous Delivery Vehicle Routing Problem (ADVRP) and present its mixed-integer linear programming formulation. Due to the NP-hardness of the problem, we propose a two-phase metaheuristic approach that first clusters customers and determines stopping locations for the ADVs, followed by a phase that determines the optimal routes for the ADVs using hybrid variable neighborhood search and simulated annealing. To evaluate our proposed solution methodology, we conduct computational experiments on various related Vehicle Routing Problems (VRPs) from the literature and newly generated ADVRP instances. The results show that the proposed two-phase metaheuristic approach can produce high-quality solutions with minimal computational effort while outperforming an exact solver in 26 medium- and large-sized instances of ADVRP and reaching optimal solutions in most VRP instances and related problems. Furthermore, we conduct sensitivity analyses on selected problem parameters and present a case study in Istanbul, Turkey to provide managerial insights for implementing ADVs in urban logistics.

Recovering feasibility in real-time conflict-free vehicle routing

Conflict-Free Vehicle Routing Problems (CF-VRPs) arise in manufacturing, transportation and logistics facilities where Automated Guided Vehicles (AGVs) are utilized to move loads. Unlike Vehicle Routing Problems arising in distribution management, CF-VRPs explicitly consider the limited capacity of the arcs of the guide path network to avoid collisions among vehicles. AGV applications have two peculiar features. First, the uncertainty affecting both travel times and machine ready times may result in vehicle delays or anticipations with respect to the fleet nominal plan. Second, the relatively high vehicle speed (in the order of one or two meters per second) requires vehicle plans to be revised in a very short amount of time (usually few milliseconds) in order to avoid collisions. In this paper we present fast exact algorithms to recover plan feasibility in real-time. In particular, we identify two corrective actions that can be implemented in real-time and formulate the problem as a linear program with the aim to optimize four common performance measures (total vehicle delay, total weighted delay, maximum route duration and total lateness). Moreover, we develop tailored algorithms which, tested on randomly generated instances of various sizes, prove to be three orders of magnitude faster than using off-the-shelf solvers.

An adaptive memory matheuristic for the set orienteering problem

This paper proposes a novel matheuristic algorithm for the Set Orienteering Problem (SOP). The set orienteering problem generalizes the Orienteering Problem (OP) by considering customers to be divided into mutually exclusive clusters. The profit associated with each cluster is collected by visiting at least one of the customers belonging to this cluster. The problem calls for the determination of the closed route that maximizes the collected profit without violating a given maximum route duration. We propose a matheuristic algorithm based on local search. The proposed algorithm is equipped with mathematical programming components for dealing with various subproblems, as well as an adaptive memory structure for producing high-quality starting solutions. Promising and diverse solutions are collected and multiple solution reconstruction mechanisms are presented. Extensive computational experiments are conducted for parameter tuning, and for evaluating the contribution of the mathematical programming components and the adaptive memory mechanism. The proposed solution approach is compared against previously published SOP algorithms. It produces the best solutions for 98.20% of the instances of the classic SOP benchmark data set. For 102 of the total 612 instances a new best solution is obtained. In addition, a new large data set of 304 instances is introduced with instances of up to 3162 customers and 634 clusters to evaluate the scalability of the proposed algorithm, and the effectiveness of the various adaptive memory schemes. The proposed algorithm manages to outperform or match the best-known solution for 281 out of 304 instances when compared with a state-of-the-art open source SOP algorithm.

A New Exact Algorithm for Single-Commodity Vehicle Routing with Split Pickups and Deliveries

We present a new exact algorithm to solve a challenging vehicle routing problem with split pickups and deliveries, named as the single-commodity split-pickup and split-delivery vehicle routing problem (SPDVRP). In the SPDVRP, any amount of a product collected from a pickup customer can be supplied to any delivery customer, and the demand of each customer can be collected or delivered multiple times by the same or different vehicles. The vehicle fleet is homogeneous with limited capacity and maximum route duration. This problem arises regularly in inventory and routing rebalancing applications, such as in bike-sharing systems, where bikes must be rebalanced over time such that the appropriate number of bikes and open docks are available to users. The solution of the SPDVRP requires determining the number of visits to each customer, the relevant portions of the demands to be collected from or delivered to the customers, and the routing of the vehicles. These three decisions are intertwined, contributing to the hardness of the problem. Our new exact algorithm for the SPDVRP is a branch-price-and-cut algorithm based on a pattern-based mathematical formulation. The SPDVRP relies on a novel label-setting algorithm used to solve the pricing problem associated with the pattern-based formulation, where the label components embed reduced cost functions, unlike those classical components that embed delivered or collected quantities, thus significantly reducing the dimension of the corresponding state space. Extensive computational results on different classes of benchmark instances illustrate that the newly proposed exact algorithm solves several open SPDVRP instances and significantly improves the running times of state-of-the-art algorithms. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72222011, 71971090, 71821001, 72171112], by the Young Elite Scientists Sponsorship Program by CAST [Grant 2019QNRC001], and by the Research Grants Council of Hong Kong SAR, China [Grant 15221619]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.1249 .

A hybrid algorithm for the multi-depot heterogeneous dial-a-ride problem

Vehicle routing problems arise in many practical situations in the context of transportation logistics. Among them, we can highlight the problem of transporting customers from origin to destination locations, which is known as the dial-a-ride problem (DARP). This problem consists of designing least-cost routes to serve pickup-and-delivery requests, while meeting capacity, time window, maximum route duration, and maximum ride time constraints. This work proposes a hybrid algorithm to solve single and multi-depot DARP variants where both the demands and vehicle fleet are heterogeneous. The method combines the iterated local search metaheuristic with an exact procedure based on a set partitioning approach. In addition, several procedures were implemented to speedup the local search phase. Extensive computational experiments were conducted on existing and newly proposed benchmark instances in order to evaluate the impact of the different components of the algorithm, and to compare its performance against the best existing method. The results obtained suggest that the proposed algorithm is capable of producing highly competitive results regarding both solution quality and CPU time, and of improving several best known solutions.

The Team Orienteering Problem with Overlaps: An Application in Cash Logistics

The team orienteering problem (TOP) aims at finding a set of routes subject to maximum route duration constraints that maximize the total collected profit from a set of customers. Motivated by a real-life automated teller machine cash replenishment problem that seeks for routes maximizing the number of bank account holders having access to cash withdrawal, we investigate a generalization of the TOP that we call the team orienteering problem with overlaps (TOPO). For this problem, the sum of individual profits may overestimate the real profit. We present exact solution methods based on column generation and a metaheuristic based on large neighborhood search to solve the TOPO. An extensive computational analysis shows that the proposed solution methods can efficiently solve synthetic and real-life TOPO instances. Moreover, the proposed methods are competitive with the best algorithms from the literature for the TOP. In particular, the exact methods can find the optimal solution of 371 of the 387 benchmark TOP instances, 33 of which are closed for the first time.

Adaptive Large Neighborhood Search with a Constant-Time Feasibility Test for the Dial-a-Ride Problem

In the dial-a-ride problem, user-specified transport requests from origin to destination points have to be served by a fleet of homogeneous vehicles. The problem variant we consider aims at finding a set of minimum-cost routes satisfying constraints on vehicle capacity, time windows, maximum route duration, and maximum user ride times. We propose an adaptive large neighborhood search (ALNS) for its solution. The key novelty of the approach is an exact amortized constant-time algorithm for evaluating the feasibility of request insertions in the repair steps of the ALNS. In addition, we use two optional improvement techniques: a local-search-based intraroute improvement of routes of promising solutions using the Balas–Simonetti neighborhood and the solution of a set covering model over a subset of all routes generated during the search. With these techniques, the proposed algorithm outperforms the state-of-the-art methods in terms of solution quality. New best solutions are found for several benchmark instances. The online appendix is available at https://doi.org/10.1287/trsc.2018.0837 .

The delivery problem: Optimizing hit rates in e-commerce deliveries

Unsuccessful delivery attempts, or failed hits, are still a recurring problem in the fulfillment of e-commerce orders to private customers. In this paper, we consider a parcel delivery company interested in optimizing the rate of successful deliveries. By doing so, the company is able to offer a differentiated service, increasing customer satisfaction, and reducing the costs related to failed delivery attempts. In order to achieve this, routes must be designed in a way that visiting times are convenient for the customers. Revisits to some customers may also be planned, so that the expected number of successful deliveries increases. We propose availability profiles to represent the availability of customers during the delivery period. Using these profiles, we are able to compute the expected number of successful hits in a given route. We model the delivery problem as a set-partitioning problem, and solve it with a branch-and-price algorithm. The corresponding pricing problem is solved with a labeling procedure, in which reduced cost bounds are employed to discard unpromising partial routes. We show that the reduced cost of route extensions is bounded by the optimal solution to an orienteering problem, and efficiently compute bounds for that problem within the labeling procedure. Computational experiments demonstrate the effectiveness of the approach for solving instances with up to 100 customers. A tradeoff analysis suggests that significant hit rate improvement can be achieved at the expense of small additional transportation cost. The results also indicate that flexibility regarding maximum route duration translates into an improved hit rate, and that planning revisits may reduce expected unsuccessful deliveries by more than 10%.

Exact loading and unloading strategies for the static multi-vehicle bike repositioning problem

This study investigates a bike repositioning problem (BRP) that determines the routes of the repositioning vehicles and the loading and unloading quantities at each bike station to firstly minimize the positive deviation from the tolerance of total demand dissatisfaction (TDD) and then service time. The total demand dissatisfaction of a bike-sharing system in this study is defined as the sum of the difference between the bike deficiency and unloading quantity of each station in the system. Two service times are considered: the total service time and the maximum route duration of the fleet. To reduce the computation time to solve the loading and unloading sub-problem of the BRP, this study examines a novel set of loading and unloading strategies and further proves them to be optimal for a given route. This set of strategies is then embedded into an enhanced artificial bee colony algorithm to solve the BRP. The numerical results demonstrate that a larger fleet size may not lead to a lower total service time but can effectively lead to a lower maximum route duration at optimality. The results also illustrate the trade-offs between the two service times, between total demand dissatisfaction and total service time, and between the number of operating vehicles provided and the TDD. Moreover, the results demonstrate that the optimal values of the two service times can increase with the TDD and introducing an upper bound on one service time can reduce the optimal value of the other service time.

A stochastic dairy transportation problem considering collection and delivery phases

Based on the dairy cooperatives in Indonesia, this study formulates a stochastic programming model to handle the milk collection and delivery process with the maximum route duration limitation, the external cooling facility option, and travel time uncertainty. Besides, the local practice of sharing the fleet for both collection and delivery further complicates the decision. A set covering-based solution approach is used, due to its flexibility in handling complicated operational requirements and alignment with the decision environment. According to the numerical experiment, the proposed solution algorithm can provide quality solutions with a reasonable computational time for the cooperative operators.

A combined multistart random constructive heuristic and set partitioning based formulation for the vehicle routing problem with time dependent travel times

Although the Vehicle Routing Problem (VRP) has been broadly addressed in the literature, most of the works consider constant travel times. This is a strong simplification that does not allow to correctly model real world applications. In fact, nowadays, travel times sensibly change, across the day, due to congestion phenomena. Therefore, to actually represent the reality, it is necessary to consider time dependent travel times. In this paper, the VRP with Time Dependent Travel Times, service times at nodes, and limit on the maximum route duration, is addressed. The objective function consists into minimizing the total travel time. A Multistart Random Constructive Heuristic, (MRCH), in which congestion level is considered, is proposed. The routes obtained by the MRCH are then used as columns in a Set Partitioning formulation. Computational results, carried out on instances derived by VRP instances taken from the literature, show the efficiency and effectiveness of the proposed approach.

The Hybrid Vehicle Routing Problem

In this paper the Hybrid Vehicle Routing Problem (HVRP) is introduced and formalized. This problem is an extension of the classical VRP in which vehicles can work both electrically and with traditional fuel. The vehicle may change propulsion mode at any point of time. The unitary travel cost is much lower for distances covered in the electric mode. An electric battery has a limited capacity and may be recharged at a recharging station (RS). A limited number of RS are available. Once a battery has been completely discharged, the vehicle automatically shifts to traditional fuel propulsion mode. Furthermore, a maximum route duration is imposed according to contracts regulations established with the driver. In this paper, a Mixed Integer Linear Programming formulation is presented and a Large Neighborhood Search based Matheuristic is proposed. The algorithm starts from a feasible solution and consists into destroying, at each iteration, a small number of routes, letting unvaried the other ones, and reconstructing a new feasible solution running the model on only the subset of customers involved in the destroyed routes. This procedure allows to completely explore a large neighborhood within very short computational time. Computational tests that show the performance of the matheuristic are presented. The method has also been tested on a simplified version of the HVRP already presented in the literature, the Green Vehicle Routing Problem (GVRP), and competitive results have been obtained.

A real-life Multi Depot Multi Period Vehicle Routing Problem with a Heterogeneous Fleet: Formulation and Adaptive Large Neighborhood Search based Matheuristic

In this paper, a new rich Vehicle Routing Problem that could arise in a real life context is introduced and formalized: the Multi Depot Multi Period Vehicle Routing Problem with a Heterogeneous Fleet. The goal of the problem is to minimize the total delivery cost. A heterogeneous fleet composed of vehicles with different capacity, characteristics (i.e. refrigerated vehicles) and hourly costs is considered. A limit on the maximum route duration is imposed. Unlike what happens in classical multi-depot VRP, not every customer may/will be served by all the vehicles or from all the depots. The planning horizon, as in most real life applications, consists of multiple periods, and the period in which each route is performed is a variable of the problem. The set of periods, within the time horizon, in which the delivery may be carried out is known for each customer. A Mixed Integer Programming (MIP) formulation for MDMPVRPHF is presented in this paper, and an Adaptive Large Neighborhood Search (ALNS) based Matheuristic approach is proposed, in which different destroy operators are defined. Computational results, pertaining to realistic instances, which show the effectiveness of the proposed method, are provided.

The Dial-a-Ride Problem with Split Requests and Profits

In this paper we introduce the dial-a-ride problem with split requests and profits (DARPSRP). Users place transportation requests, specifying a pickup location, a delivery location, and a time window for either of the two. Based on maximum user ride time considerations, the second time window is generated. A given fleet of vehicles, each with a certain capacity, is available to serve these requests, and maximum route duration constraints have to be respected. Each request is associated with a revenue and the objective is to maximize the total profit, that is, the total revenue minus the total costs. Transportation requests involving several persons may be split if it is beneficial to do so. We formulate the DARPSRP as a mixed-integer program using position variables and in terms of a path-based formulation. For the solution of the latter, we design a branch-and-price algorithm. The largest instance solved to optimality, when applied to available instances from the literature, has 40 requests; when applied to newly generated instances, the largest instance solved to optimality consists of 24 requests. To solve larger instances a variable neighborhood search algorithm is developed. We investigate the impact of request splitting under different geographical settings, assuming favorable settings for request splitting in terms of the number of people per request. The largest benefits from request splitting are obtained for problem settings exhibiting clustered customer locations.

A matheuristic for the Team Orienteering Arc Routing Problem

In the Team Orienteering Arc Routing Problem (TOARP) the potential customers are located on the arcs of a directed graph and are to be chosen on the basis of an associated profit. A limited fleet of vehicles is available to serve the chosen customers. Each vehicle has to satisfy a maximum route duration constraint. The goal is to maximize the profit of the served customers. We propose a matheuristic for the TOARP and test it on a set of benchmark instances for which the optimal solution or an upper bound is known. The matheuristic finds the optimal solutions on all, except one, instances of one of the four classes of tested instances (with up to 27 vertices and 296 arcs). The average error on all instances for which the optimal solution is available is 0.67 percent.

A Two-Phase Scheduling Method Combined to the Tabu Search for the DARP

The dial-a-ride problem (DARP), is a variant of the pickup and delivery problem (PDP), consists of designing vehicle routes of n customers transportation requests. The problem arises in many transportation applications, like door-to-door transportation services for elderly and disabled people or in services for patients. This paper consider a static multivehicle DARP, which the objective is to minimize a combined costs of total travel distance, total duration, passengers waiting time, the excess ride time of customers, and the early arrival time while respecting maximum route duration limit, the maximum costumer ride time limit, the capacity and the time window constraint. The authors propose a two-phase scheduling method combined to the tabu search heuristic, for the static multivehicle DARP. Their experimentation report best results for Cordeau Benchmark test problem, compared to reported results.

Transit Network Design: a Hybrid Enhanced Artificial Bee Colony Approach and a Case Study

A bus network design problem in a suburban area of Hong Kong is studied. The objective is to minimize the weighted sum of the number of transfers and the total travel time of passengers by restructuring bus routes and determining new frequencies. A mixed integer optimization model is developed and was solved by a Hybrid Enhanced Artificial Bee Colony algorithm (HEABC). A case study was conducted to investigate the effects of different design parameters, including the total number of bus routes available, the maximum route duration within the study area and the maximum allowable number of bus routes that originated from each terminal. The model and results are useful for improving bus service policies.

Dial-a-Ride Problem with time windows, transshipments, and dynamic transfer points

Dial-a-ride problems deal with on demand transportation. Today, this type of shared transport is used for elderly and disabled people for short distances. People provide a request for transport from an origin to a destination, both specific for a particular demand. Time constraints deal with maximum user ride time, time windows, maximum route duration limits and precedence. In this paper we try to solve these dial-a-ride problems, including the possibility of one transshipment from a dynamic transfer point by request. We propose an algorithm based on insertion techniques and constraints propagation.